module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.CategoryTheory.Sites.IsSheafFor | {
"line": 1013,
"column": 2
} | {
"line": 1014,
"column": 51
} | {
"line": 1015,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX : C\nP : Cᵒᵖ ⥤ Type u_1\nU : Sieve X\nB : ⦃Y : C⦄ → ⦃f : Y ⟶ X⦄ → U.arrows f → Sieve Y\nhU : IsSheafFor P U.arrows\nhB : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (hf : U.arrows f), IsSheafFor P (B hf).arrows\nhB' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (h : U.arrows f) ⦃Z : C⦄ (g : Z ⟶ Y), IsS... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX : C\nP : Cᵒᵖ ⥤ Type u_1\nU : Sieve X\nB : ⦃Y : C⦄ → ⦃f : Y ⟶ X⦄ → U.arrows f → Sieve Y\nhU : IsSheafFor P U.arrows\nhB : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (hf : U.arrows f), IsSheafFor P (B hf).arrows\nhB' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (h : U.arrows f) ⦃Z : C⦄ (g : Z ⟶ Y), IsSeparatedFor ... | let t : Presieve.FamilyOfElements P (U : Presieve X) :=
fun Y f hf => (hB hf).amalgamate (y hf) (hy hf) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.CategoryTheory.Sites.Subsheaf | {
"line": 139,
"column": 76
} | {
"line": 142,
"column": 43
} | {
"line": 144,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nF : Cᵒᵖ ⥤ Type w\nG : Subfunctor F\nh : Presieve.IsSheaf J F\n⊢ Presieve.IsSheaf J G.toFunctor ↔ ∀ (U : Cᵒᵖ) (s : F.obj U), G.sieveOfSection s ∈ J (unop U) → s ∈ G.obj U",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants"... | [] | by
rw [← G.eq_sheafify_iff h]
change _ ↔ G.sheafify J ≤ G
exact ⟨Eq.ge, (G.le_sheafify J).antisymm⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Subsheaf | {
"line": 297,
"column": 8
} | {
"line": 297,
"column": 36
} | {
"line": 298,
"column": 8
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nF✝ F'✝ F'' : Cᵒᵖ ⥤ Type w\nG G' : Subfunctor F✝\nF F' : Sheaf J (Type (max v u))\nf : F ⟶ F'\nI : Limits.MonoFactorisation f\nM : Mono I.m.hom\n⊢ Subfunctor.sheafify J (Subfunctor.range f.hom) ≤ Subfunctor.range I.m.hom",
"ppTerm": ... | [
"case h\nC : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nF✝ F'✝ F'' : Cᵒᵖ ⥤ Type w\nG G' : Subfunctor F✝\nF F' : Sheaf J (Type (max v u))\nf : F ⟶ F'\nI : Limits.MonoFactorisation f\nM : Mono I.m.hom\n⊢ Subfunctor.range f.hom ≤ Subfunctor.range I.m.hom",
"case hF\nC : Type u\ninst✝ : Category.{... | apply Subfunctor.sheafify_le | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Category.TopCat.Limits.Pullbacks | {
"line": 158,
"column": 4
} | {
"line": 158,
"column": 17
} | {
"line": 159,
"column": 4
} | [
{
"pp": "J : Type v\ninst✝³ : Category.{w, v} J\nX✝ Y✝ Z✝ : TopCat\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf : X → Z\nhf : Continuous[inst✝², inst✝] f\ng : Y → Z\nhg : IsEmbedding g\nx : { p // f p.1 = g p.2 }\n⊢ (fun x ↦ ⟨... | [
"case snd\nJ : Type v\ninst✝³ : Category.{w, v} J\nX✝ Y✝ Z✝ : TopCat\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf : X → Z\nhf : Continuous[inst✝², inst✝] f\ng : Y → Z\nhg : IsEmbedding g\nx : { p // f p.1 = g p.2 }\n⊢ Exists.choos... | ext <;> dsimp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Topology.Category.TopCat.Limits.Products | {
"line": 307,
"column": 10
} | {
"line": 309,
"column": 33
} | {
"line": 310,
"column": 6
} | [
{
"pp": "X Y : TopCat\nc : BinaryCofan X Y\nh₁ : IsOpenEmbedding ⇑(ConcreteCategory.hom c.inl)\nh₂ : IsOpenEmbedding ⇑(ConcreteCategory.hom c.inr)\nh₃ : IsCompl (range ⇑(ConcreteCategory.hom c.inl)) (range ⇑(ConcreteCategory.hom c.inr))\nthis :\n ∀ (x : ↑(((Functor.const (Discrete WalkingPair)).obj c.pt).obj {... | [] | · change IsOpen (Set.range c.inl)ᶜ
rw [← eq_compl_iff_isCompl.mpr h₃.symm]
exact h₂.isOpen_range | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.ObjectProperty.Local | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 72
} | {
"line": 70,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nW W' : MorphismProperty C\nZ Z' : C\ne : Z ≅ Z'\nhZ : W.isLocal Z\nX Y : C\nf : X ⟶ Y\nhf : W f\n⊢ Function.Bijective fun g ↦ f ≫ g",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"CategoryTheory.Iso.homToEquiv",
"Eq.mpr",
"... | [
"C : Type u\ninst✝ : Category.{v, u} C\nW W' : MorphismProperty C\nZ Z' : C\ne : Z ≅ Z'\nhZ : W.isLocal Z\nX Y : C\nf : X ⟶ Y\nhf : W f\n⊢ Function.Bijective ((fun g ↦ f ≫ g) ∘ ⇑e.homToEquiv)"
] | rw [← Function.Bijective.of_comp_iff _ (Iso.homToEquiv e).bijective] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.ObjectProperty.Local | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 9
} | {
"line": 73,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nW W' : MorphismProperty C\nZ Z' : C\ne : Z ≅ Z'\nhZ : W.isLocal Z\nX Y : C\nf : X ⟶ Y\nhf : W f\n⊢ (fun g ↦ f ≫ g) ∘ ⇑e.homToEquiv = ⇑e.homToEquiv ∘ fun g ↦ f ≫ g",
"ppTerm": "?m.101",
"assigned": true,
"usedConstants": [
"CategoryTheory.Iso.homT... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.ObjectProperty.Local | {
"line": 77,
"column": 4
} | {
"line": 77,
"column": 9
} | {
"line": 79,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nW W' : MorphismProperty C\nX X' : C\ne : X ≅ X'\nhX : W.isColocal X\nY Z : C\ng : Y ⟶ Z\nhg : W g\n⊢ (fun f ↦ f ≫ g) ∘ ⇑e.homFromEquiv = ⇑e.homFromEquiv ∘ fun f ↦ f ≫ g",
"ppTerm": "?m.101",
"assigned": true,
"usedConstants": [
"CategoryTheory.Ca... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.ObjectProperty.Local | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 7
} | {
"line": 114,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nW W' : MorphismProperty C\nh : W ≤ W'\nf : C\nhf : W'.isLocal f\n⊢ W.isLocal f",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"_private.Mathlib.CategoryTheory.ObjectProper... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.ObjectProperty.Local | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 7
} | {
"line": 121,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nW W' : MorphismProperty C\nh : W ≤ W'\nf : C\nhf : W'.isColocal f\n⊢ W.isColocal f",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty.isColocal",
"CategoryTheory.CategoryStruct.toQuive... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.ObjectProperty.Local | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 7
} | {
"line": 127,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nι : Sort u_1\nW : ι → MorphismProperty C\n⊢ (⨆ i, W i).isLocal = ⨅ i, (W i).isLocal",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"iInf",
"CategoryTheory.CategoryStruct.to... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.ObjectProperty.Local | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 7
} | {
"line": 127,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nι : Sort u_1\nW : ι → MorphismProperty C\n⊢ (⨆ i, W i).isLocal = ⨅ i, (W i).isLocal",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"iInf",
"CategoryTheory.CategoryStruct.to... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.ObjectProperty.Local | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 7
} | {
"line": 127,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nι : Sort u_1\nW : ι → MorphismProperty C\n⊢ (⨆ i, W i).isLocal = ⨅ i, (W i).isLocal",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"iInf",
"CategoryTheory.CategoryStruct.to... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.ObjectProperty.Local | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 7
} | {
"line": 133,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nι : Sort u_1\nW : ι → MorphismProperty C\n⊢ (⨆ i, W i).isColocal = ⨅ i, (W i).isColocal",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"iInf",
"CategoryTheory.MorphismPrope... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.ObjectProperty.Local | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 7
} | {
"line": 133,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nι : Sort u_1\nW : ι → MorphismProperty C\n⊢ (⨆ i, W i).isColocal = ⨅ i, (W i).isColocal",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"iInf",
"CategoryTheory.MorphismPrope... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.ObjectProperty.Local | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 7
} | {
"line": 133,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nι : Sort u_1\nW : ι → MorphismProperty C\n⊢ (⨆ i, W i).isColocal = ⨅ i, (W i).isColocal",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"iInf",
"CategoryTheory.MorphismPrope... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Adhesive.Basic | {
"line": 331,
"column": 34
} | {
"line": 422,
"column": 27
} | {
"line": 424,
"column": 0
} | [
{
"pp": "J : Type v'\ninst✝⁴ : Category.{u', v'} J\nC : Type u\ninst✝³ : Category.{v, u} C\nW X Y Z✝ : C\nf✝ : W ⟶ X\ng✝ : W ⟶ Y\nh : X ⟶ Z✝\ni : Y ⟶ Z✝\ninst✝² : Adhesive C\nZ A B : C\na : A ⟶ Z\nb : B ⟶ Z\ninst✝¹ : Mono a\ninst✝ : Mono b\nK : C\nf g : K ⟶ pushout (fst a b) (snd a b)\nw : f ≫ pushout.desc a b ... | [] | by
/- First, take the pullback of `a` and `b` and then form the pushout of the projection maps:
`pullback a b` -> `B`
| |
| `v`
| |
v v
`A` ---`u`---> C -/
let u := pushout.inl (pullback.fst a b) (pullback.snd a... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Localization | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 15
} | {
"line": 107,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nJ : GrothendieckTopology C\ninst✝ : LocallySmall.{w, v_1, u_1} C\nX : C\nS : Sieve X\nhS : S ∈ J X\nZ : Cᵒᵖ ⥤ Type w\nhZ : Presieve.IsSheaf J Z\n⊢ Presieve.IsSheafFor Z S.arrows",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [],
"us... | [] | exact hZ _ hS | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Localization.Bousfield | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 7
} | {
"line": 224,
"column": 0
} | [
{
"pp": "case e'_3\nC : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\nF : C ⥤ D\nG : D ⥤ C\nadj : G ⊣ F\ninst✝¹ : F.Full\ninst✝ : F.Faithful\nX : D\nY : C\ne_1✝ : ((G ⋙ F).obj X ⟶ F.obj Y) = (F.obj (G.obj X) ⟶ F.obj Y)\ne_2✝ : ((𝟭 D).obj X ⟶ F.obj Y) = (X ⟶ F.obj Y)\n⊢... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.Sites.LocallyInjective | {
"line": 54,
"column": 33
} | {
"line": 54,
"column": 38
} | {
"line": 56,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\nFD : D → D → Type u_1\nCD : D → Type w\ninst✝¹ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)\ninst✝ : ConcreteCategory D FD\nF : Cᵒᵖ ⥤ D\nX : Cᵒᵖ\nx : ToType (F.obj X)\n⊢ equalizerSieve x x = ⊤",
"ppTerm": "?m.32",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.Sites.LocallyInjective | {
"line": 54,
"column": 33
} | {
"line": 54,
"column": 38
} | {
"line": 56,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\nFD : D → D → Type u_1\nCD : D → Type w\ninst✝¹ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)\ninst✝ : ConcreteCategory D FD\nF : Cᵒᵖ ⥤ D\nX : Cᵒᵖ\nx : ToType (F.obj X)\n⊢ equalizerSieve x x = ⊤",
"ppTerm": "?m.32",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.LocallyInjective | {
"line": 54,
"column": 33
} | {
"line": 54,
"column": 38
} | {
"line": 56,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\nFD : D → D → Type u_1\nCD : D → Type w\ninst✝¹ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)\ninst✝ : ConcreteCategory D FD\nF : Cᵒᵖ ⥤ D\nX : Cᵒᵖ\nx : ToType (F.obj X)\n⊢ equalizerSieve x x = ⊤",
"ppTerm": "?m.32",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.LocallyInjective | {
"line": 174,
"column": 4
} | {
"line": 174,
"column": 25
} | {
"line": 175,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\nFD : D → D → Type u_1\nCD : D → Type w\ninst✝¹ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)\ninst✝ : ConcreteCategory D FD\nJ : GrothendieckTopology C\nF₁ F₂ F₃ : Cᵒᵖ ⥤ D\nφ : F₁ ⟶ F₂\nψ : F₂ ⟶ F₃\nF : Cᵒᵖ ⥤ Type w\nG : ... | [
"C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\nFD : D → D → Type u_1\nCD : D → Type w\ninst✝¹ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)\ninst✝ : ConcreteCategory D FD\nJ : GrothendieckTopology C\nF₁ F₂ F₃ : Cᵒᵖ ⥤ D\nφ : F₁ ⟶ F₂\nψ : F₂ ⟶ F₃\nF : Cᵒᵖ ⥤ Type w\nG : Subfunctor F... | intro ⟨x, _⟩ ⟨y, _⟩ h | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Sites.LocallySurjective | {
"line": 172,
"column": 4
} | {
"line": 172,
"column": 62
} | {
"line": 173,
"column": 2
} | [
{
"pp": "case mp\nC : Type u\ninst✝⁴ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝³ : Category.{v', u'} A\nFA : A → A → Type u_1\nCA : A → Type w'\ninst✝² : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)\ninst✝¹ : ConcreteCategory A FA\nF₁ F₂ F₃ : Cᵒᵖ ⥤ A\nf₁ : F₁ ⟶ F₂\nf₂ : F₂ ⟶ F₃\nf₃ : F₁ ⟶... | [] | exact isLocallySurjective_of_isLocallySurjective_fac J fac | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Sites.LocallySurjective | {
"line": 256,
"column": 4
} | {
"line": 257,
"column": 82
} | {
"line": 258,
"column": 2
} | [
{
"pp": "case mp\nC : Type u\ninst✝⁵ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝⁴ : Category.{v', u'} A\nFA : A → A → Type u_1\nCA : A → Type w'\ninst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)\ninst✝² : ConcreteCategory A FA\nF₁ F₂ F₃ : Cᵒᵖ ⥤ A\nf₁ : F₁ ⟶ F₂\nf₂ : F₂ ⟶ F₃\ninst✝¹ : ... | [] | intro
exact isLocallySurjective_of_isLocallySurjective_of_isLocallyInjective J f₁ f₂ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.LocallySurjective | {
"line": 256,
"column": 4
} | {
"line": 257,
"column": 82
} | {
"line": 258,
"column": 2
} | [
{
"pp": "case mp\nC : Type u\ninst✝⁵ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝⁴ : Category.{v', u'} A\nFA : A → A → Type u_1\nCA : A → Type w'\ninst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)\ninst✝² : ConcreteCategory A FA\nF₁ F₂ F₃ : Cᵒᵖ ⥤ A\nf₁ : F₁ ⟶ F₂\nf₂ : F₂ ⟶ F₃\ninst✝¹ : ... | [] | intro
exact isLocallySurjective_of_isLocallySurjective_of_isLocallyInjective J f₁ f₂ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.LocallySurjective | {
"line": 424,
"column": 4
} | {
"line": 434,
"column": 75
} | {
"line": 435,
"column": 2
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝² : Category.{v, u} C\nS : C\nι : Type u_2\ninst✝¹ : Small.{w, u_2} ι\nX : ι → C\nf : (i : ι) → X i ⟶ S\ninst✝ : LocallySmall.{w, v, u} C\nc : Cofan fun i ↦ shrinkYoneda.{w, v, u}.obj (X i)\nhc : IsColimit c\nU : C\ng : U ⟶ S\nV : C\nv : V ⟶ U\nhv :\n (imageSieve (Cofan... | [] | obtain ⟨w, hw⟩ := hv
obtain ⟨⟨i⟩, a, rfl⟩ := Types.jointly_surjective_of_isColimit
(isColimitOfPreserves ((evaluation _ _).obj (op V)) hc) w
obtain ⟨a : V ⟶ X i, rfl⟩ := shrinkYonedaObjObjEquiv.symm.surjective a
refine ⟨_, a, _, ⟨i⟩, shrinkYonedaObjObjEquiv.symm.injective ?_⟩
rw [← shrinkYoneda_ma... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.LocallySurjective | {
"line": 424,
"column": 4
} | {
"line": 434,
"column": 75
} | {
"line": 435,
"column": 2
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝² : Category.{v, u} C\nS : C\nι : Type u_2\ninst✝¹ : Small.{w, u_2} ι\nX : ι → C\nf : (i : ι) → X i ⟶ S\ninst✝ : LocallySmall.{w, v, u} C\nc : Cofan fun i ↦ shrinkYoneda.{w, v, u}.obj (X i)\nhc : IsColimit c\nU : C\ng : U ⟶ S\nV : C\nv : V ⟶ U\nhv :\n (imageSieve (Cofan... | [] | obtain ⟨w, hw⟩ := hv
obtain ⟨⟨i⟩, a, rfl⟩ := Types.jointly_surjective_of_isColimit
(isColimitOfPreserves ((evaluation _ _).obj (op V)) hc) w
obtain ⟨a : V ⟶ X i, rfl⟩ := shrinkYonedaObjObjEquiv.symm.surjective a
refine ⟨_, a, _, ⟨i⟩, shrinkYonedaObjObjEquiv.symm.injective ?_⟩
rw [← shrinkYoneda_ma... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafify | {
"line": 73,
"column": 35
} | {
"line": 73,
"column": 40
} | {
"line": 73,
"column": 41
} | [
{
"pp": "case a\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrpCat\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nY :... | [
"case a\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrpCat\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nY : C\nr₀ r₀' :... | hg.1, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.Projective.Dimension | {
"line": 146,
"column": 32
} | {
"line": 146,
"column": 47
} | {
"line": 146,
"column": 47
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\nX : C\ninst✝ : HasExt C\nh : ∀ ⦃Y : C⦄, Subsingleton (Ext X Y 1)\nE✝ X✝ : C\nf : X ⟶ X✝\ng : E✝ ⟶ X✝\nx✝ : Epi g\n⊢ (Ext.mk₀ f).comp ⋯.extClass ⋯ = 0",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"CategoryTheory.A... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Preserves.Opposites | {
"line": 529,
"column": 17
} | {
"line": 531,
"column": 70
} | {
"line": 533,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : Cᵒᵖ ⥤ Dᵒᵖ\ninst✝ : PreservesFiniteProducts F\nx✝ : ℕ\n⊢ PreservesColimitsOfShape (Discrete (Fin x✝)) F.unop",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Opposite",
"CategoryTh... | [] | by
apply +allowSynthFailures preservesColimitsOfShape_unop
exact preservesLimitsOfShape_of_equiv (Discrete.opposite _).symm _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Functor.Flat | {
"line": 212,
"column": 34
} | {
"line": 212,
"column": 45
} | {
"line": 212,
"column": 45
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nJ : Type v₁\ninst✝² : SmallCategory J\ninst✝¹ : FinCategory J\nF : C ⥤ D\ninst✝ : RepresentablyFlat F\nK : J ⥤ C\nc : Cone K\nhc : IsLimit c\ns : Cone (K ⋙ F)\nf₁ f₂ : s.pt ⟶ F.obj c.pt\nh₁ : ∀ (j : J), f₁ ≫ (F.mapCon... | [] | simp [← h₂] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Functor.Flat | {
"line": 212,
"column": 34
} | {
"line": 212,
"column": 45
} | {
"line": 212,
"column": 45
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nJ : Type v₁\ninst✝² : SmallCategory J\ninst✝¹ : FinCategory J\nF : C ⥤ D\ninst✝ : RepresentablyFlat F\nK : J ⥤ C\nc : Cone K\nhc : IsLimit c\ns : Cone (K ⋙ F)\nf₁ f₂ : s.pt ⟶ F.obj c.pt\nh₁ : ∀ (j : J), f₁ ≫ (F.mapCon... | [] | simp [← h₂] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Functor.Flat | {
"line": 212,
"column": 34
} | {
"line": 212,
"column": 45
} | {
"line": 212,
"column": 45
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nJ : Type v₁\ninst✝² : SmallCategory J\ninst✝¹ : FinCategory J\nF : C ⥤ D\ninst✝ : RepresentablyFlat F\nK : J ⥤ C\nc : Cone K\nhc : IsLimit c\ns : Cone (K ⋙ F)\nf₁ f₂ : s.pt ⟶ F.obj c.pt\nh₁ : ∀ (j : J), f₁ ≫ (F.mapCon... | [] | simp [← h₂] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Functor.Flat | {
"line": 238,
"column": 6
} | {
"line": 239,
"column": 10
} | {
"line": 240,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nJ : Type v₁\ninst✝² : SmallCategory J\ninst✝¹ : FinCategory J\nF : C ⥤ D\ninst✝ : RepresentablyFlat F\nK : J ⥤ C\nc : Cone K\nhc : IsLimit c\ns : Cone (K ⋙ F)\nf₁ f₂ : s.pt ⟶ F.obj c.pt\nh₁ : ∀ (j : J), f₁ ≫ (F.mapCon... | [] | apply hc.uniq (c.extend _)
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Functor.Flat | {
"line": 238,
"column": 6
} | {
"line": 239,
"column": 10
} | {
"line": 240,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nJ : Type v₁\ninst✝² : SmallCategory J\ninst✝¹ : FinCategory J\nF : C ⥤ D\ninst✝ : RepresentablyFlat F\nK : J ⥤ C\nc : Cone K\nhc : IsLimit c\ns : Cone (K ⋙ F)\nf₁ f₂ : s.pt ⟶ F.obj c.pt\nh₁ : ∀ (j : J), f₁ ≫ (F.mapCon... | [] | apply hc.uniq (c.extend _)
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Bicones | {
"line": 131,
"column": 27
} | {
"line": 131,
"column": 37
} | {
"line": 131,
"column": 38
} | [
{
"pp": "J : Type v₁\ninst✝¹ : SmallCategory J\ninst✝ : FinCategory J\nf : Bicone.left ⟶ Bicone.right\n⊢ f ∈ ∅",
"ppTerm": "?m.233",
"assigned": true,
"usedConstants": [
"CategoryTheory.Bicone.right",
"CategoryTheory.Bicone.ctorIdx",
"CategoryTheory.Bicone.diagram",
"Category... | [] | by cases f | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Bicones | {
"line": 137,
"column": 27
} | {
"line": 137,
"column": 37
} | {
"line": 137,
"column": 38
} | [
{
"pp": "J : Type v₁\ninst✝¹ : SmallCategory J\ninst✝ : FinCategory J\nf : Bicone.right ⟶ Bicone.left\n⊢ f ∈ ∅",
"ppTerm": "?m.363",
"assigned": true,
"usedConstants": [
"CategoryTheory.Bicone.right",
"CategoryTheory.Bicone.ctorIdx",
"CategoryTheory.Bicone.diagram",
"Category... | [] | by cases f | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Bicones | {
"line": 146,
"column": 27
} | {
"line": 146,
"column": 37
} | {
"line": 146,
"column": 38
} | [
{
"pp": "J : Type v₁\ninst✝¹ : SmallCategory J\ninst✝ : FinCategory J\nval✝ : J\nf : Bicone.diagram val✝ ⟶ Bicone.left\n⊢ f ∈ ∅",
"ppTerm": "?m.559",
"assigned": true,
"usedConstants": [
"CategoryTheory.Bicone.right",
"CategoryTheory.Bicone.ctorIdx",
"CategoryTheory.Bicone.diagram"... | [] | by cases f | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Bicones | {
"line": 149,
"column": 27
} | {
"line": 149,
"column": 37
} | {
"line": 149,
"column": 38
} | [
{
"pp": "J : Type v₁\ninst✝¹ : SmallCategory J\ninst✝ : FinCategory J\nval✝ : J\nf : Bicone.diagram val✝ ⟶ Bicone.right\n⊢ f ∈ ∅",
"ppTerm": "?m.621",
"assigned": true,
"usedConstants": [
"CategoryTheory.Bicone.right",
"CategoryTheory.Bicone.ctorIdx",
"CategoryTheory.Bicone.diagram... | [] | by cases f | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Coverage | {
"line": 413,
"column": 6
} | {
"line": 413,
"column": 40
} | {
"line": 414,
"column": 6
} | [
{
"pp": "case mpr.H\nC : Type u_2\ninst✝ : Category.{v_1, u_2} C\nK : Coverage C\nP : Cᵒᵖ ⥤ Type u_1\nH : ∀ {X : C}, ∀ R ∈ K.coverings X, IsSheafFor P R\nX Y : C\nf : Y ⟶ X\nS : Presieve X\nhS : S ∈ K.coverings X\nT : Presieve Y\nhT1 : T ∈ K.coverings Y\nhT2 : T.FactorsThruAlong S f\nZ : C\ng : Z ⟶ Y\nhg : T g\... | [
"case mpr.H\nC : Type u_2\ninst✝ : Category.{v_1, u_2} C\nK : Coverage C\nP : Cᵒᵖ ⥤ Type u_1\nH : ∀ {X : C}, ∀ R ∈ K.coverings X, IsSheafFor P R\nX Y : C\nf : Y ⟶ X\nS : Presieve X\nhS : S ∈ K.coverings X\nT : Presieve Y\nhT1 : T ∈ K.coverings Y\nhT2 : T.FactorsThruAlong S f\nZ : C\ng : Z ⟶ Y\nhg : T g\nW : C\ni : ... | obtain ⟨W, i, e, h1, h2⟩ := hT2 hg | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Sites.Coverage | {
"line": 470,
"column": 2
} | {
"line": 470,
"column": 7
} | {
"line": 472,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Category.{v_2, u_2} D\nK : Coverage C\nP : Cᵒᵖ ⥤ D\n⊢ (∀ (E : D) {X : C}, ∀ R ∈ K.coverings X, Presieve.IsSheafFor (P ⋙ coyoneda.obj (Opposite.op E)) R) ↔\n ∀ ⦃X : C⦄, ∀ R ∈ K.coverings X, ∀ (E : Dᵒᵖ), Presieve.IsSheafFor (P ⋙ coyon... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck | {
"line": 221,
"column": 42
} | {
"line": 221,
"column": 76
} | {
"line": 221,
"column": 76
} | [
{
"pp": "C : Type u_3\ninst✝ : Category.{u_2, u_3} C\nF : Cᵒᵖ ⥤ Type u_1\nS : C\n𝒰 𝒱 : PreZeroHypercover S\ne : 𝒰 ≅ 𝒱\n⊢ Presieve.IsSheafFor F 𝒱.sieve₀.arrows ↔ Presieve.IsSheafFor F 𝒱.presieve₀",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Pre... | [
"C : Type u_3\ninst✝ : Category.{u_2, u_3} C\nF : Cᵒᵖ ⥤ Type u_1\nS : C\n𝒰 𝒱 : PreZeroHypercover S\ne : 𝒰 ≅ 𝒱\n⊢ Presieve.IsSheafFor F (Presieve.ofArrows 𝒱.X 𝒱.f) ↔ Presieve.IsSheafFor F 𝒱.presieve₀"
] | ← Presieve.isSheafFor_iff_generate | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.Continuous | {
"line": 268,
"column": 2
} | {
"line": 269,
"column": 44
} | {
"line": 271,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst✝ : F.IsContinuous J K\nG : Dᵒᵖ ⥤ Type u_1\nh : Presieve.IsSheaf K G\n⊢ Presieve.IsSheaf J (F.op ⋙ G)",
"ppTerm": "?m.49",
"assigned": tru... | [] | rw [← isSheaf_iff_isSheaf_of_type] at h ⊢
exact F.op_comp_isSheaf_of_isSheaf _ _ _ h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Continuous | {
"line": 268,
"column": 2
} | {
"line": 269,
"column": 44
} | {
"line": 271,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst✝ : F.IsContinuous J K\nG : Dᵒᵖ ⥤ Type u_1\nh : Presieve.IsSheaf K G\n⊢ Presieve.IsSheaf J (F.op ⋙ G)",
"ppTerm": "?m.49",
"assigned": tru... | [] | rw [← isSheaf_iff_isSheaf_of_type] at h ⊢
exact F.op_comp_isSheaf_of_isSheaf _ _ _ h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.CoverLifting | {
"line": 239,
"column": 2
} | {
"line": 245,
"column": 80
} | {
"line": 247,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\nG : C ⥤ D\nA : Type w\ninst✝¹ : Category.{w', w} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst✝ : G.IsCocontinuous J K\nF : Cᵒᵖ ⥤ A\nhF : Presheaf.IsSheaf J F\nR : Dᵒᵖ ⥤ A\nα : G.op ⋙ R ⟶ F\nhR... | [] | apply (hR (op i.Y)).hom_ext
intro j
have eq := fac' hF hR s (StructuredArrow.mk (i.f.op ≫ j.hom))
dsimp at eq ⊢
simp only [Functor.map_comp, Category.assoc] at eq
rw [Category.assoc, eq]
simpa using liftAux_map hF α s (j.hom.unop ≫ i.f) (𝟙 _) i j.hom.unop (by simp) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.CoverLifting | {
"line": 239,
"column": 2
} | {
"line": 245,
"column": 80
} | {
"line": 247,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\nG : C ⥤ D\nA : Type w\ninst✝¹ : Category.{w', w} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst✝ : G.IsCocontinuous J K\nF : Cᵒᵖ ⥤ A\nhF : Presheaf.IsSheaf J F\nR : Dᵒᵖ ⥤ A\nα : G.op ⋙ R ⟶ F\nhR... | [] | apply (hR (op i.Y)).hom_ext
intro j
have eq := fac' hF hR s (StructuredArrow.mk (i.f.op ≫ j.hom))
dsimp at eq ⊢
simp only [Functor.map_comp, Category.assoc] at eq
rw [Category.assoc, eq]
simpa using liftAux_map hF α s (j.hom.unop ≫ i.f) (𝟙 _) i j.hom.unop (by simp) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.CoverPreserving | {
"line": 164,
"column": 4
} | {
"line": 167,
"column": 34
} | {
"line": 168,
"column": 4
} | [
{
"pp": "case hex\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nhF₁ : CompatiblePreserving K F\nhF₂ : CoverPreserving J K F\nG : Sheaf K (Type (max u₁ v₁ u₂ v₂))\nX : C\nS : Sieve X\nhS : S ∈ J X\nx : Fami... | [
"case hunique\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nhF₁ : CompatiblePreserving K F\nhF₂ : CoverPreserving J K F\nG : Sheaf K (Type (max u₁ v₁ u₂ v₂))\nX : C\nS : Sieve X\nhS : S ∈ J X\nx : FamilyOfElem... | · have H := (isSheaf_iff_isSheaf_of_type _ _).1 G.2 _ (hF₂.cover_preserve hS)
exact ⟨H.amalgamate (x.functorPushforward F) (hx.functorPushforward hF₁),
fun V f hf => (H.isAmalgamation (hx.functorPushforward hF₁) (F.map f) _).trans
(hF₁.apply_map _ hx hf)⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.CoverPreserving | {
"line": 183,
"column": 24
} | {
"line": 183,
"column": 58
} | {
"line": 183,
"column": 58
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst✝ : F.IsContinuous J K\nX : C\nι : Type (max u₁ v₁)\nY : ι → C\nf : (i : ι) → Y i ⟶ X\nhS : Sieve.ofArrows Y f ∈ J X\nthis : IsSheafFor (F.op ⋙ (F... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst✝ : F.IsContinuous J K\nX : C\nι : Type (max u₁ v₁)\nY : ι → C\nf : (i : ι) → Y i ⟶ X\nhS : Sieve.ofArrows Y f ∈ J X\nthis : IsSheafFor (F.op ⋙ (Functor.close... | ← Presieve.isSheafFor_iff_generate | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.LocallyFullyFaithful | {
"line": 63,
"column": 24
} | {
"line": 63,
"column": 29
} | {
"line": 65,
"column": 0
} | [
{
"pp": "C : Type uC\ninst✝¹ : Category.{vC, uC} C\nD : Type uD\ninst✝ : Category.{vD, uD} D\nG : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nU V : C\nf₁ f₂ : U ⟶ V\n⊢ ∀ {Y Z : C} {f : Y ⟶ U}, f ≫ f₁ = f ≫ f₂ → ∀ (g : Z ⟶ Y), (g ≫ f) ≫ f₁ = (g ≫ f) ≫ f₂",
"ppTerm": "?m.28",
"assigned"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.Sites.LocallyFullyFaithful | {
"line": 63,
"column": 24
} | {
"line": 63,
"column": 29
} | {
"line": 65,
"column": 0
} | [
{
"pp": "C : Type uC\ninst✝¹ : Category.{vC, uC} C\nD : Type uD\ninst✝ : Category.{vD, uD} D\nG : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nU V : C\nf₁ f₂ : U ⟶ V\n⊢ ∀ {Y Z : C} {f : Y ⟶ U}, f ≫ f₁ = f ≫ f₂ → ∀ (g : Z ⟶ Y), (g ≫ f) ≫ f₁ = (g ≫ f) ≫ f₂",
"ppTerm": "?m.28",
"assigned"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.LocallyFullyFaithful | {
"line": 63,
"column": 24
} | {
"line": 63,
"column": 29
} | {
"line": 65,
"column": 0
} | [
{
"pp": "C : Type uC\ninst✝¹ : Category.{vC, uC} C\nD : Type uD\ninst✝ : Category.{vD, uD} D\nG : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nU V : C\nf₁ f₂ : U ⟶ V\n⊢ ∀ {Y Z : C} {f : Y ⟶ U}, f ≫ f₁ = f ≫ f₂ → ∀ (g : Z ⟶ Y), (g ≫ f) ≫ f₁ = (g ≫ f) ≫ f₂",
"ppTerm": "?m.28",
"assigned"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.DenseSubsite.Basic | {
"line": 599,
"column": 6
} | {
"line": 601,
"column": 51
} | {
"line": 601,
"column": 51
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝² : Category.{v_2, u_2} D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nA : Type u_4\ninst✝¹ : Category.{v_4, u_4} A\ninst✝ : IsDenseSubsite J K G\nF : Sheaf J A\nX Y : C\nf₁ f₂ : X ⟶ Y\nh : G.map f₁ = G.map f₂\n⊢ ∀ ... | [] | rintro ⟨W₀, a, ha⟩
dsimp at ha ⊢
simp only [← Functor.map_comp, ← op_comp, ha] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.DenseSubsite.Basic | {
"line": 599,
"column": 6
} | {
"line": 601,
"column": 51
} | {
"line": 601,
"column": 51
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝² : Category.{v_2, u_2} D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nA : Type u_4\ninst✝¹ : Category.{v_4, u_4} A\ninst✝ : IsDenseSubsite J K G\nF : Sheaf J A\nX Y : C\nf₁ f₂ : X ⟶ Y\nh : G.map f₁ = G.map f₂\n⊢ ∀ ... | [] | rintro ⟨W₀, a, ha⟩
dsimp at ha ⊢
simp only [← Functor.map_comp, ← op_comp, ha] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.DenseSubsite.Basic | {
"line": 772,
"column": 23
} | {
"line": 778,
"column": 71
} | {
"line": 780,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝⁵ : Category.{v_2, u_2} D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nA : Type u_4\ninst✝⁴ : Category.{v_4, u_4} A\ninst✝³ : IsDenseSubsite J K G\ninst✝² : (G.sheafPushforwardContinuous A J K).IsEquivalence\ninst✝¹... | [] | by
have : PreservesFiniteLimits (presheafToSheaf J A ⋙
(G.sheafPushforwardContinuous A J K).inv) := by
apply comp_preservesFiniteLimits
have : PreservesFiniteLimits (sheafifyOfIsEquivalence J K G A) := by
apply comp_preservesFiniteLimits
exact HasSheafify.mk' _ _ (sheafifyAdjunctionOfIsEquivalence J K... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Equivalence | {
"line": 89,
"column": 6
} | {
"line": 89,
"column": 77
} | {
"line": 90,
"column": 4
} | [
{
"pp": "case mp\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nK : GrothendieckTopology D\ne : C ≌ D\ninst✝¹ : e.functor.IsCocontinuous J K\ninst✝ : e.inverse.IsCocontinuous K J\nX : C\nS : Sieve X\nH : Sieve.functorPushforward e.functor S ∈ K... | [] | rw [(Sieve.fullyFaithfulFunctorGaloisCoinsertion e.functor X).u_l_eq S] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.CoversTop.Basic | {
"line": 150,
"column": 2
} | {
"line": 151,
"column": 61
} | {
"line": 153,
"column": 0
} | [
{
"pp": "case hunique\nC : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nF : Cᵒᵖ ⥤ Type w\nI : Type u_1\nY : I → C\nx : FamilyOfElementsOnObjects F Y\nhx : x.IsCompatible\nhY : J.CoversTop Y\nhF : IsSheaf J F\nH : Presieve.IsSheaf J F\n⊢ ∀ (y₁ y₂ : ↑F.sections),\n (∀ (i : I), ↑y₁ (Opposite.o... | [] | · intro y₁ y₂ hy₁ hy₂
exact hY.sections_ext ⟨F, hF⟩ (fun i => by rw [hy₁, hy₂]) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.Over | {
"line": 118,
"column": 35
} | {
"line": 118,
"column": 42
} | {
"line": 118,
"column": 42
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : C\nY₁ Y₂ : Over X\nf : Y₁ ⟶ Y₂\nS : Sieve Y₂\nZ : C\ng : Z ⟶ Y₁.left\nW : Over X\na : W ⟶ Y₂\nb : Z ⟶ W.left\nh : S.arrows a\nw : g ≫ Over.Hom.left f = b ≫ Over.Hom.left a\nT : Over X := ⋯\nc : T ⟶ Y₁ := ⋯\nd : T ⟶ W := ⋯\n⊢ Over.Hom.left (c ≫ f) = Over.Hom.le... | [] | exact w | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Sites.Over | {
"line": 132,
"column": 68
} | {
"line": 134,
"column": 6
} | {
"line": 136,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : C\nY : Over X\nS : Sieve Y\nZ : C\nf : Z ⟶ Y.left\n⊢ ((overEquiv Y) S).arrows f ↔ S.arrows (Over.homMk f ⋯)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"CategoryTheory.Over",
"CategoryTheory.Sieve.overEquiv_iff._proof_1",
... | [] | by
obtain ⟨S, rfl⟩ := (overEquiv Y).symm.surjective S
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.CoversTop.Over | {
"line": 32,
"column": 91
} | {
"line": 32,
"column": 96
} | {
"line": 32,
"column": 96
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nJ : GrothendieckTopology C\nI : Type u_2\nX : I → C\nhX : J.CoversTop X\nI' : I → Type u\nY : (i : I) → I' i → Over (X i)\nhY : ∀ (i : I), (J.over (X i)).CoversTop (Y i)\nj Z : C\nf : Z ⟶ j\nx✝ : (Sieve.ofObjects X j).arrows f\ni : I\ng : Z ⟶ X i\ni' : C\n⊢ ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.Sites.CoversTop.Over | {
"line": 32,
"column": 91
} | {
"line": 32,
"column": 96
} | {
"line": 32,
"column": 96
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nJ : GrothendieckTopology C\nI : Type u_2\nX : I → C\nhX : J.CoversTop X\nI' : I → Type u\nY : (i : I) → I' i → Over (X i)\nhY : ∀ (i : I), (J.over (X i)).CoversTop (Y i)\nj Z : C\nf : Z ⟶ j\nx✝ : (Sieve.ofObjects X j).arrows f\ni : I\ng : Z ⟶ X i\ni' : C\n⊢ ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.CoversTop.Over | {
"line": 32,
"column": 91
} | {
"line": 32,
"column": 96
} | {
"line": 32,
"column": 96
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nJ : GrothendieckTopology C\nI : Type u_2\nX : I → C\nhX : J.CoversTop X\nI' : I → Type u\nY : (i : I) → I' i → Over (X i)\nhY : ∀ (i : I), (J.over (X i)).CoversTop (Y i)\nj Z : C\nf : Z ⟶ j\nx✝ : (Sieve.ofObjects X j).arrows f\ni : I\ng : Z ⟶ X i\ni' : C\n⊢ ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Over | {
"line": 428,
"column": 2
} | {
"line": 428,
"column": 7
} | {
"line": 430,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝ : Category.{v', u'} A\nX Y : C\nf g : X ⟶ Y\nh : f = g\n⊢ J.overMapPullbackCongr A h = eqToIso ⋯",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"CategoryTheory.Over.map",
"CategoryT... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.Sites.Over | {
"line": 428,
"column": 2
} | {
"line": 428,
"column": 7
} | {
"line": 430,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝ : Category.{v', u'} A\nX Y : C\nf g : X ⟶ Y\nh : f = g\n⊢ J.overMapPullbackCongr A h = eqToIso ⋯",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"CategoryTheory.Over.map",
"CategoryT... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Over | {
"line": 428,
"column": 2
} | {
"line": 428,
"column": 7
} | {
"line": 430,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝ : Category.{v', u'} A\nX Y : C\nf g : X ⟶ Y\nh : f = g\n⊢ J.overMapPullbackCongr A h = eqToIso ⋯",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"CategoryTheory.Over.map",
"CategoryT... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sheaves.SheafCondition.Sites | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 58
} | {
"line": 65,
"column": 2
} | [
{
"pp": "case a\nX : TopCat\nU : Opens ↑X\nR : Presieve U\nhR : Sieve.generate R ∈ (Opens.grothendieckTopology ↑X) U\nx : ↑X\nhxU : x ∈ U\n⊢ ∃ i, x ∈ coveringOfPresieve U R i",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | [
"case a\nX : TopCat\nU : Opens ↑X\nR : Presieve U\nhR : Sieve.generate R ∈ (Opens.grothendieckTopology ↑X) U\nx : ↑X\nhxU : x ∈ U\nV : Opens ↑X\niVU : V ⟶ U\nhxV : x ∈ V\nW : Opens ↑X\niVW : V ⟶ W\niWU : W ⟶ U\nhiWU : R iWU\n⊢ ∃ i, x ∈ coveringOfPresieve U R i"
] | obtain ⟨V, iVU, ⟨W, iVW, iWU, hiWU, -⟩, hxV⟩ := hR x hxU | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing | {
"line": 103,
"column": 4
} | {
"line": 103,
"column": 23
} | {
"line": 104,
"column": 2
} | [
{
"pp": "case single.pair.right\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\nFC : C → C → Type u_2\nCC : C → Type u_3\ninst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝ : ConcreteCategory C FC\nX : TopCat\nF : Presheaf (Type u_4) X\nι : Type u_5\nU : ι → Opens ↑X\nsf : (i : ι) → ToType (F.obj (op (U ... | [] | exact (h i' i).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing | {
"line": 103,
"column": 4
} | {
"line": 103,
"column": 23
} | {
"line": 104,
"column": 2
} | [
{
"pp": "case single.pair.right\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\nFC : C → C → Type u_2\nCC : C → Type u_3\ninst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝ : ConcreteCategory C FC\nX : TopCat\nF : Presheaf (Type u_4) X\nι : Type u_5\nU : ι → Opens ↑X\nsf : (i : ι) → ToType (F.obj (op (U ... | [] | exact (h i' i).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing | {
"line": 103,
"column": 4
} | {
"line": 103,
"column": 23
} | {
"line": 104,
"column": 2
} | [
{
"pp": "case single.pair.right\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\nFC : C → C → Type u_2\nCC : C → Type u_3\ninst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝ : ConcreteCategory C FC\nX : TopCat\nF : Presheaf (Type u_4) X\nι : Type u_5\nU : ι → Opens ↑X\nsf : (i : ι) → ToType (F.obj (op (U ... | [] | exact (h i' i).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Sheaf.Quasicoherent | {
"line": 282,
"column": 4
} | {
"line": 283,
"column": 48
} | {
"line": 285,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR : Sheaf J RingCat\ninst✝² : ∀ (X : C), HasWeakSheafify (J.over X) AddCommGrpCat\ninst✝¹ : ∀ (X : C), (J.over X).WEqualsLocallyBijective AddCommGrpCat\nM : SheafOfModules R\ninst✝ : M.IsFinitePresentation\n⊢ ∃ σ, σ.IsFiniteType",
... | [] | obtain ⟨σ, _⟩ := IsFinitePresentation.exists_quasicoherentData M
exact ⟨σ.localGeneratorsData, inferInstance⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Sheaf.Quasicoherent | {
"line": 282,
"column": 4
} | {
"line": 283,
"column": 48
} | {
"line": 285,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR : Sheaf J RingCat\ninst✝² : ∀ (X : C), HasWeakSheafify (J.over X) AddCommGrpCat\ninst✝¹ : ∀ (X : C), (J.over X).WEqualsLocallyBijective AddCommGrpCat\nM : SheafOfModules R\ninst✝ : M.IsFinitePresentation\n⊢ ∃ σ, σ.IsFiniteType",
... | [] | obtain ⟨σ, _⟩ := IsFinitePresentation.exists_quasicoherentData M
exact ⟨σ.localGeneratorsData, inferInstance⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Module.Basic | {
"line": 68,
"column": 6
} | {
"line": 68,
"column": 31
} | {
"line": 68,
"column": 31
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁷ : Ring R\ninst✝⁶ : TopologicalSpace R\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : AddCommGroup M\ninst✝³ : ContinuousAdd M\ninst✝² : Module R M\ninst✝¹ : ContinuousSMul R M\ninst✝ : (𝓝[{x | IsUnit x}] 0).NeBot\ns : Submodule R M\ny : M\nhy : y ∈ interior ↑s\nx : M\n⊢ x ∈ ... | [
"R : Type u_1\nM : Type u_2\ninst✝⁷ : Ring R\ninst✝⁶ : TopologicalSpace R\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : AddCommGroup M\ninst✝³ : ContinuousAdd M\ninst✝² : Module R M\ninst✝¹ : ContinuousSMul R M\ninst✝ : (𝓝[{x | IsUnit x}] 0).NeBot\ns : Submodule R M\ny : M\nhy : ↑s ∈ 𝓝 y\nx : M\n⊢ x ∈ s"
] | mem_interior_iff_mem_nhds | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Sheaves.Stalks | {
"line": 477,
"column": 95
} | {
"line": 480,
"column": 98
} | {
"line": 482,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasColimits C\nX : TopCat\nFC : C → C → Type u_1\nCC : C → Type v\ninst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninstCC : ConcreteCategory C FC\ninst✝ : PreservesFilteredColimits (forget C)\nB : Set (Opens ↑X)\nhB : Opens.IsBasis B\nF : Presheaf C... | [] | by
obtain ⟨U, hxU, s, rfl⟩ := F.exists_germ_eq t
obtain ⟨_, ⟨V, hV, rfl⟩, hxV, hVU⟩ := hB.exists_subset_of_mem_open hxU U.2
exact ⟨V, hxV, hV, F.map (homOfLE hVU).op s, by rw [← ConcreteCategory.comp_apply, F.germ_res']⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Module.ModuleTopology | {
"line": 551,
"column": 22
} | {
"line": 551,
"column": 27
} | {
"line": 551,
"column": 27
} | [
{
"pp": "case fst\nR : Type u_1\ninst✝⁹ : TopologicalSpace R\ninst✝⁸ : Semiring R\nM : Type u_2\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : IsModuleTopology R M\nN : Type u_3\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : TopologicalSpace N\ninst✝ : IsModuleTo... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Algebra.Module.ModuleTopology | {
"line": 551,
"column": 22
} | {
"line": 551,
"column": 27
} | {
"line": 551,
"column": 27
} | [
{
"pp": "case snd\nR : Type u_1\ninst✝⁹ : TopologicalSpace R\ninst✝⁸ : Semiring R\nM : Type u_2\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : IsModuleTopology R M\nN : Type u_3\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : TopologicalSpace N\ninst✝ : IsModuleTo... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Algebra.Module.Equiv | {
"line": 665,
"column": 2
} | {
"line": 665,
"column": 7
} | {
"line": 667,
"column": 0
} | [
{
"pp": "R₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\ninst✝¹⁵ : Semiring R₁\ninst✝¹⁴ : Semiring R₂\ninst✝¹³ : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₁ : R₂ →+* R₁\ninst✝¹² : RingHomInvPair σ₁₂ σ₂₁\ninst✝¹¹ : RingHomInvPair σ₂₁ σ₁₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\ninst✝¹⁰ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\nM₁ : Type u_... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Algebra.Module.Equiv | {
"line": 665,
"column": 2
} | {
"line": 665,
"column": 7
} | {
"line": 667,
"column": 0
} | [
{
"pp": "R₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\ninst✝¹⁵ : Semiring R₁\ninst✝¹⁴ : Semiring R₂\ninst✝¹³ : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₁ : R₂ →+* R₁\ninst✝¹² : RingHomInvPair σ₁₂ σ₂₁\ninst✝¹¹ : RingHomInvPair σ₂₁ σ₁₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\ninst✝¹⁰ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\nM₁ : Type u_... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Module.Equiv | {
"line": 665,
"column": 2
} | {
"line": 665,
"column": 7
} | {
"line": 667,
"column": 0
} | [
{
"pp": "R₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\ninst✝¹⁵ : Semiring R₁\ninst✝¹⁴ : Semiring R₂\ninst✝¹³ : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₁ : R₂ →+* R₁\ninst✝¹² : RingHomInvPair σ₁₂ σ₂₁\ninst✝¹¹ : RingHomInvPair σ₂₁ σ₁₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\ninst✝¹⁰ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\nM₁ : Type u_... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Module.Equiv | {
"line": 670,
"column": 2
} | {
"line": 670,
"column": 7
} | {
"line": 672,
"column": 0
} | [
{
"pp": "R₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\ninst✝¹⁵ : Semiring R₁\ninst✝¹⁴ : Semiring R₂\ninst✝¹³ : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₁ : R₂ →+* R₁\ninst✝¹² : RingHomInvPair σ₁₂ σ₂₁\ninst✝¹¹ : RingHomInvPair σ₂₁ σ₁₂\nσ₃₂ : R₃ →+* R₂\nσ₃₁ : R₃ →+* R₁\ninst✝¹⁰ : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁\nM₁ : Type u_... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Algebra.Module.Equiv | {
"line": 670,
"column": 2
} | {
"line": 670,
"column": 7
} | {
"line": 672,
"column": 0
} | [
{
"pp": "R₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\ninst✝¹⁵ : Semiring R₁\ninst✝¹⁴ : Semiring R₂\ninst✝¹³ : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₁ : R₂ →+* R₁\ninst✝¹² : RingHomInvPair σ₁₂ σ₂₁\ninst✝¹¹ : RingHomInvPair σ₂₁ σ₁₂\nσ₃₂ : R₃ →+* R₂\nσ₃₁ : R₃ →+* R₁\ninst✝¹⁰ : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁\nM₁ : Type u_... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Module.Equiv | {
"line": 670,
"column": 2
} | {
"line": 670,
"column": 7
} | {
"line": 672,
"column": 0
} | [
{
"pp": "R₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\ninst✝¹⁵ : Semiring R₁\ninst✝¹⁴ : Semiring R₂\ninst✝¹³ : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₁ : R₂ →+* R₁\ninst✝¹² : RingHomInvPair σ₁₂ σ₂₁\ninst✝¹¹ : RingHomInvPair σ₂₁ σ₁₂\nσ₃₂ : R₃ →+* R₂\nσ₃₁ : R₃ →+* R₁\ninst✝¹⁰ : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁\nM₁ : Type u_... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Topology.Basic | {
"line": 121,
"column": 7
} | {
"line": 121,
"column": 38
} | {
"line": 121,
"column": 38
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\ninst✝ : TopologicalSpace R\nX Y : TopModuleCat R\ne : ↑X.toModuleCat ≃L[R] ↑Y.toModuleCat\n⊢ ofHom ↑e ≫ ofHom ↑e.symm = 𝟙 X",
"ppTerm": "?m.105",
"assigned": true,
"usedConstants": [
"TopModuleCat.instCategory",
"ContinuousLinearEquiv.symm",
"... | [] | ext; exact e.symm_apply_apply _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Topology.Basic | {
"line": 121,
"column": 7
} | {
"line": 121,
"column": 38
} | {
"line": 121,
"column": 38
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\ninst✝ : TopologicalSpace R\nX Y : TopModuleCat R\ne : ↑X.toModuleCat ≃L[R] ↑Y.toModuleCat\n⊢ ofHom ↑e ≫ ofHom ↑e.symm = 𝟙 X",
"ppTerm": "?m.105",
"assigned": true,
"usedConstants": [
"TopModuleCat.instCategory",
"ContinuousLinearEquiv.symm",
"... | [] | ext; exact e.symm_apply_apply _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Topology.Basic | {
"line": 465,
"column": 33
} | {
"line": 467,
"column": 18
} | {
"line": 469,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\ninst✝ : TopologicalSpace R\nX : TopModuleCat R\n⊢ { app := fun X ↦ TopCat.ofHom { toFun := fun x ↦ Finsupp.single x 1, continuous_toFun := ⋯ }, naturality := ⋯ }.app\n ((forget₂ (TopModuleCat R) TopCat).obj X) ≫\n (forget₂ (TopModuleCat R) TopCat).map\n (... | [] | by
ext
simp [freeObj] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.FiniteMultiequalizer | {
"line": 67,
"column": 10
} | {
"line": 67,
"column": 69
} | {
"line": 68,
"column": 10
} | [
{
"pp": "case pos\nJ : MultispanShape\ninst✝³ : Fintype J.L\ninst✝² : Fintype J.R\ninst✝¹ : DecidableEq J.L\ninst✝ : DecidableEq J.R\na : J.L\nb : J.R\nh₁ : J.fst a = b\nh₂ : J.snd a = b\nf : (left a ⟶ right b) → Prop :=\n failed to pretty print expression (use 'set_option pp.rawOnError true' for raw represent... | [
"case pos\nJ : MultispanShape\ninst✝³ : Fintype J.L\ninst✝² : Fintype J.R\ninst✝¹ : DecidableEq J.L\ninst✝ : DecidableEq J.R\na : J.L\nb : J.R\nh₁ : J.fst a = b\nh₂ : J.snd a = b\nf : (left a ⟶ right b) → Prop :=\n failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\n⊢ Tr... | conv_rhs => tactic => subst h₂; simp only [eqToHom_refl, f] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convRHS_1 | Mathlib.Tactic.Conv.convRHS |
Mathlib.CategoryTheory.Presentable.IsCardinalFiltered | {
"line": 190,
"column": 87
} | {
"line": 190,
"column": 92
} | {
"line": 191,
"column": 4
} | [
{
"pp": "κ : Cardinal.{w}\nhκ : Fact κ.IsRegular\nι : Type w\nf : ι → κ.ord.ToType\nhs : Cardinal.mk ι < κ\nx✝ : { x // x ∈ Set.range f }\n⊢ ∃ a, (fun i ↦ ⟨f i, ⋯⟩) a = x✝",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Subtype.casesOn",
"Memb... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.Presentable.IsCardinalFiltered | {
"line": 190,
"column": 87
} | {
"line": 190,
"column": 92
} | {
"line": 191,
"column": 4
} | [
{
"pp": "κ : Cardinal.{w}\nhκ : Fact κ.IsRegular\nι : Type w\nf : ι → κ.ord.ToType\nhs : Cardinal.mk ι < κ\nx✝ : { x // x ∈ Set.range f }\n⊢ ∃ a, (fun i ↦ ⟨f i, ⋯⟩) a = x✝",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Subtype.casesOn",
"Memb... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Presentable.IsCardinalFiltered | {
"line": 190,
"column": 87
} | {
"line": 190,
"column": 92
} | {
"line": 191,
"column": 4
} | [
{
"pp": "κ : Cardinal.{w}\nhκ : Fact κ.IsRegular\nι : Type w\nf : ι → κ.ord.ToType\nhs : Cardinal.mk ι < κ\nx✝ : { x // x ∈ Set.range f }\n⊢ ∃ a, (fun i ↦ ⟨f i, ⋯⟩) a = x✝",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Subtype.casesOn",
"Memb... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.MorphismProperty | {
"line": 128,
"column": 4
} | {
"line": 128,
"column": 31
} | {
"line": 130,
"column": 0
} | [
{
"pp": "T : Type u_1\ninst✝⁵ : Category.{v_1, u_1} T\nA : Type u_2\ninst✝⁴ : Category.{v_2, u_2} A\nL : A ⥤ T\nJ : Type u_3\ninst✝³ : Category.{v_3, u_3} J\nP : MorphismProperty T\ninst✝² : P.RespectsIso\ninst✝¹ : PreservesColimitsOfShape J L\ninst✝ : HasColimitsOfShape J A\nc : (D : J ⥤ T) → [HasColimit D] → ... | [] | exact H _ _ d.prop_diag_obj | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Limits.MorphismProperty | {
"line": 210,
"column": 4
} | {
"line": 210,
"column": 31
} | {
"line": 212,
"column": 0
} | [
{
"pp": "T : Type u_1\ninst✝⁵ : Category.{v_1, u_1} T\nA : Type u_2\ninst✝⁴ : Category.{v_2, u_2} A\nL : A ⥤ T\nJ : Type u_3\ninst✝³ : Category.{v_3, u_3} J\nP : MorphismProperty T\ninst✝² : P.RespectsIso\ninst✝¹ : PreservesLimitsOfShape J L\ninst✝ : HasLimitsOfShape J A\nc : (D : J ⥤ T) → [HasLimit D] → Cone D... | [] | exact H _ _ d.prop_diag_obj | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.CharP.MixedCharZero | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 60
} | {
"line": 99,
"column": 4
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝ : CommRing R\nP : Prop\nh : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P\nq : ℕ\nq_pos : q > 0\nq_mixedChar : MixedCharZero R q\n⊢ P",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"CommSemiring.toSemiring",
"Ad... | [
"case mpr\nR : Type u_1\ninst✝ : CommRing R\nP : Prop\nh : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P\nq : ℕ\nq_pos : q > 0\nq_mixedChar : MixedCharZero R q\nI : Ideal R\nhI_ne_top : I ≠ ⊤\nright✝ : CharP (R ⧸ I) q\n⊢ P"
] | rcases q_mixedChar.charP_quotient with ⟨I, hI_ne_top, _⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Algebra.CharP.MixedCharZero | {
"line": 293,
"column": 58
} | {
"line": 296,
"column": 45
} | {
"line": 298,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\n⊢ IsEmpty (Algebra ℚ R) ↔ ∃ p > 0, MixedCharZero R p",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"Eq.mpr",
"Mathlib.Tactic.Push.not_and_eq",
"Semiring.toMod... | [] | by
contrapose!
rw [← EqualCharZero.iff_not_mixedCharZero]
apply EqualCharZero.nonempty_algebraRat_iff | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.CharZero.Quotient | {
"line": 40,
"column": 6
} | {
"line": 41,
"column": 65
} | {
"line": 42,
"column": 4
} | [
{
"pp": "case mp.refine_1\nR : Type u_1\ninst✝¹ : DivisionRing R\ninst✝ : CharZero R\np r : R\nz : ℤ\nhz : z ≠ 0\nhz' : ↑z ≠ 0\nk : ℤ\nh : k • p = z • r\n⊢ (k % z).toNat < z.natAbs",
"ppTerm": "?mp.refine_1",
"assigned": true,
"usedConstants": [
"Int.emod_lt_abs",
"Eq.mpr",
"abs",
... | [] | rw [← Int.ofNat_lt, Int.toNat_of_nonneg (Int.emod_nonneg _ hz)]
exact (Int.emod_lt_abs _ hz).trans_eq (Int.abs_eq_natAbs _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.CharZero.Quotient | {
"line": 40,
"column": 6
} | {
"line": 41,
"column": 65
} | {
"line": 42,
"column": 4
} | [
{
"pp": "case mp.refine_1\nR : Type u_1\ninst✝¹ : DivisionRing R\ninst✝ : CharZero R\np r : R\nz : ℤ\nhz : z ≠ 0\nhz' : ↑z ≠ 0\nk : ℤ\nh : k • p = z • r\n⊢ (k % z).toNat < z.natAbs",
"ppTerm": "?mp.refine_1",
"assigned": true,
"usedConstants": [
"Int.emod_lt_abs",
"Eq.mpr",
"abs",
... | [] | rw [← Int.ofNat_lt, Int.toNat_of_nonneg (Int.emod_nonneg _ hz)]
exact (Int.emod_lt_abs _ hz).trans_eq (Int.abs_eq_natAbs _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Stream.Init | {
"line": 54,
"column": 2
} | {
"line": 54,
"column": 19
} | {
"line": 55,
"column": 2
} | [
{
"pp": "α : Type u\nn m : ℕ\ns : Stream' α\n⊢ (drop m s).get n = s.get (m + n)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Stream'.drop",
"id",
"instHAdd",
"Stream'.get",
"HAdd.hAdd",
"Nat",
"instAddNat",
... | [
"α : Type u\nn m : ℕ\ns : Stream' α\n⊢ (drop m s).get n = s.get (n + m)"
] | rw [Nat.add_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Stream.Init | {
"line": 241,
"column": 6
} | {
"line": 241,
"column": 14
} | {
"line": 241,
"column": 14
} | [
{
"pp": "α : Type u\nf : α → α\na : α\nn : ℕ\n⊢ (iterate f a).tail.get n = (iterate f (f a)).get n",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Stream'.get_tail",
"id",
"instOfNatNat",
"instHAdd",
"Stream'.iterate",
... | [
"α : Type u\nf : α → α\na : α\nn : ℕ\n⊢ (iterate f a).get (n + 1) = (iterate f (f a)).get n"
] | get_tail | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Seq.Defs | {
"line": 378,
"column": 8
} | {
"line": 378,
"column": 20
} | {
"line": 379,
"column": 8
} | [
{
"pp": "case nil.cons\nα : Type u\nR : Seq α → Seq α → Prop\nbisim : IsBisimulation R\ns₁ s₂ : Seq α\nr : R s₁ s₂\nt₁ t₂ : Stream' (Option α)\ne : ∃ s s', ↑s = t₁ ∧ ↑s' = t₂ ∧ R s s'\nx✝ : α\ns✝ : Seq α\n⊢ R nil (cons x✝ s✝) →\n BisimO R nil.destruct (cons x✝ s✝).destruct → nil.head = (cons x✝ s✝).head ∧ R ... | [
"case nil.cons\nα : Type u\nR : Seq α → Seq α → Prop\nbisim : IsBisimulation R\ns₁ s₂ : Seq α\nr : R s₁ s₂\nt₁ t₂ : Stream' (Option α)\ne : ∃ s s', ↑s = t₁ ∧ ↑s' = t₂ ∧ R s s'\nx✝ : α\ns✝ : Seq α\nr✝ : R nil (cons x✝ s✝)\nthis : BisimO R nil.destruct (cons x✝ s✝).destruct\n⊢ nil.head = (cons x✝ s✝).head ∧ R nil.tai... | intro _ this | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Data.Seq.Defs | {
"line": 381,
"column": 8
} | {
"line": 381,
"column": 20
} | {
"line": 382,
"column": 8
} | [
{
"pp": "case cons.nil\nα : Type u\nR : Seq α → Seq α → Prop\nbisim : IsBisimulation R\ns₁ s₂ : Seq α\nr : R s₁ s₂\nt₁ t₂ : Stream' (Option α)\ne : ∃ s s', ↑s = t₁ ∧ ↑s' = t₂ ∧ R s s'\nx✝ : α\ns✝ : Seq α\n⊢ R (cons x✝ s✝) nil →\n BisimO R (cons x✝ s✝).destruct nil.destruct → (cons x✝ s✝).head = nil.head ∧ R ... | [
"case cons.nil\nα : Type u\nR : Seq α → Seq α → Prop\nbisim : IsBisimulation R\ns₁ s₂ : Seq α\nr : R s₁ s₂\nt₁ t₂ : Stream' (Option α)\ne : ∃ s s', ↑s = t₁ ∧ ↑s' = t₂ ∧ R s s'\nx✝ : α\ns✝ : Seq α\nr✝ : R (cons x✝ s✝) nil\nthis : BisimO R (cons x✝ s✝).destruct nil.destruct\n⊢ (cons x✝ s✝).head = nil.head ∧ R (cons x... | intro _ this | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Data.Seq.Defs | {
"line": 450,
"column": 65
} | {
"line": 450,
"column": 79
} | {
"line": 450,
"column": 80
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\ns : Seq α\nn : ℕ\n⊢ s.get? n = none ↔ (s.get? n).isNone = true",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Option.isNone",
"Option.casesOn",
"Option.some",
"Bool.true",
"Option.none",
"Iff",
"Bool",
... | [
"case none\nα : Type u\nβ : Type v\nγ : Type w\ns : Seq α\nn : ℕ\n⊢ none = none ↔ none.isNone = true",
"case some\nα : Type u\nβ : Type v\nγ : Type w\ns : Seq α\nn : ℕ\nval✝ : α\n⊢ some val✝ = none ↔ (some val✝).isNone = true"
] | cases s.get? n | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Data.Stream.Init | {
"line": 520,
"column": 78
} | {
"line": 520,
"column": 86
} | {
"line": 520,
"column": 86
} | [
{
"pp": "α : Type u\ns : Stream' α\nn : ℕ\n⊢ take n.succ s ++ [s.tail.get n] = take (n + 1) s ++ [s.get (n + 1)]",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Stream'.take",
"Eq.mpr",
"congrArg",
"Stream'.get_tail",
"id",
"instOfNatNat",
"List.c... | [
"α : Type u\ns : Stream' α\nn : ℕ\n⊢ take n.succ s ++ [s.get (n + 1)] = take (n + 1) s ++ [s.get (n + 1)]"
] | get_tail | Lean.Elab.Tactic.evalRewriteSeq | null |
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